Problem: Factor the following expression: $8$ $x^2+$ $31$ $x$ $-4$
This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(8)}{(-4)} &=& -32 \\ {a} + {b} &=& & & {31} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-32$ and add them together. Remember, since $-32$ is negative, one of the factors must be negative. The factors that add up to ${31}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-1}$ and ${b}$ is ${32}$ $ \begin{eqnarray} {ab} &=& ({-1})({32}) &=& -32 \\ {a} + {b} &=& {-1} + {32} &=& 31 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {8}x^2 {-1}x +{32}x {-4} $ Group the terms so that there is a common factor in each group: $ ({8}x^2 {-1}x) + ({32}x {-4}) $ Factor out the common factors: $ x(8x - 1) + 4(8x - 1) $ Notice how $(8x - 1)$ has become a common factor. Factor this out to find the answer. $(8x - 1)(x + 4)$